"""Parse and analyze finite operations represented by Markdown tables."""

from __future__ import annotations

import json


def _cells(line: str) -> list[str]:
    return [cell.strip() for cell in line.strip().strip("|").split("|")]


def analyze_markdown_operation(text: str) -> str:
    """Compute core algebraic properties of a Markdown Cayley table."""
    rows = [_cells(line) for line in text.splitlines() if line.strip().startswith("|")]
    rows = [row for row in rows if not all(set(cell) <= {"-", ":"} for cell in row)]
    if len(rows) < 2 or len(rows[0]) < 2:
        raise ValueError("No Markdown operation table found")
    elements = rows[0][1:]
    mapping: dict[tuple[str, str], str] = {}
    for row in rows[1:]:
        if len(row) >= len(elements) + 1:
            mapping.update({(row[0], b): value for b, value in zip(elements, row[1:])})
    complete = len(mapping) == len(elements) ** 2
    commutative = complete and all(
        mapping[a, b] == mapping[b, a] for a in elements for b in elements
    )
    identities = [
        e
        for e in elements
        if complete and all(mapping[e, a] == a and mapping[a, e] == a for a in elements)
    ]
    associative = complete and all(
        mapping.get((mapping[a, b], c)) == mapping.get((a, mapping[b, c]))
        for a in elements
        for b in elements
        for c in elements
    )
    noncommuting = [
        [a, b]
        for i, a in enumerate(elements)
        for b in elements[i + 1 :]
        if complete and mapping[a, b] != mapping[b, a]
    ]
    violating_elements = sorted({item for pair in noncommuting for item in pair})
    return json.dumps(
        {
            "elements": elements,
            "complete": complete,
            "commutative": commutative,
            "associative": associative,
            "identities": identities,
            "noncommuting_unordered_pairs": noncommuting,
            "elements_participating_in_violations": violating_elements,
        },
        ensure_ascii=False,
    )


def solve_markdown_question(text: str) -> str | None:
    """Return an exact deterministic answer for recognized operation-table questions."""
    analysis = json.loads(analyze_markdown_operation(text))
    lowered = text.lower()
    if (
        "do not commute" in lowered
        or "fail to commute" in lowered
        or "noncommut" in lowered
        or ("counter-example" in lowered and "commutative" in lowered)
        or ("counterexample" in lowered and "commutative" in lowered)
    ):
        return ", ".join(analysis["elements_participating_in_violations"])
    if "commutative" in lowered and ("is " in lowered or "whether" in lowered):
        return "Yes" if analysis["commutative"] else "No"
    if "associative" in lowered and ("is " in lowered or "whether" in lowered):
        return "Yes" if analysis["associative"] else "No"
    if "identity" in lowered:
        return ", ".join(analysis["identities"])
    return None